
theorem
  for T being non empty TopSpace, A being Subset of T, p being Point of
T holds p in Fr A iff ex B being Basis of p st for U being Subset of T st U in
  B holds A meets U & U \ A <> {}
proof
  let T be non empty TopSpace, A be Subset of T, p be Point of T;
  hereby
    set B = the Basis of p;
    assume
A1: p in Fr A;
    take B;
    let U be Subset of T;
    assume U in B;
    then U is open & p in U by YELLOW_8:12;
    hence A meets U & U \ A <> {} by A1,Th9;
  end;
  given B being Basis of p such that
A2: for U being Subset of T st U in B holds A meets U & U \ A <> {};
  for U being Subset of T st U is open & p in U holds A meets U & U meets A`
  proof
    let U be Subset of T;
    assume U is open & p in U;
    then consider V being Subset of T such that
A3: V in B and
A4: V c= U by YELLOW_8:def 1;
    V \ A <> {} by A2,A3;
    then
A5: U \ A <> {} by A4,XBOOLE_1:3,33;
    A meets V by A2,A3;
    hence thesis by A4,A5,Th1,XBOOLE_1:63;
  end;
  hence thesis by TOPS_1:28;
end;
